\begin{answer}
    We have
    $$
    \begin{aligned}
        D_{KL}(P(X, Y)\|Q(X, Y)) &= \sum_{y, x} P(x, y)\log\frac{P(x, y)}{Q(x, y)}\\
        &= \sum_{x, y} P(x, y) \log \frac{P(y|x)P(x)}{Q(y|x)Q(x)} \\
        & = \sum_{x,y}P(x, y)\log\frac{P(x)}{Q(x)} + \sum_{x, y}P(y|x)P(x)\log\frac{P(y|x)}{Q(y|x)}\\
        &= \sum_{x}P(x)\log\frac{P(x)}{Q(x)} + \sum_{x}P(x)\sum_{y}P(y|x)\log\frac{P(y|x)}{Q(y|x)}\\
        &= D_{KL}(P(X)\|Q(X)) + D_{KL}(P(Y|X)\|Q(Y|X))
    \end{aligned}
    $$
\end{answer}
